Resolutions of singularities.

Yoshifumi Tsuchimoto

DEFINITION 04.1   An subvariety $V=V$ of an affine space $\mathbb{A}^N$ of codimension $k$ is non-singular at its point $P$ if $V$ is locally defined by $k$ polynomials $f_1,\dots, f_k$ such that $d f_1,\dots d f_k$ is linearly independent at $P$ .

The dimension and codimensions are defined by using the transcendence degree of the extension of the function field $k(V)=Q(k[V])$ . The definition of regularity (singularity) is more naturally defined by using theory of commutative algebras.